N. Ramanujam

A numerical algorithm for singular perturbation problems exhibiting weak boundary layers

Abstract A class of singularly perturbed two-point boundary-value problems (BVP) for second-order ordinary differential equations is considered here. To avoid the numerical difficulties in the solution to these problems, we divide the domain into two subdomains. The first BVP is a layer domain problem and the second BVP is a regular domain problem. Error estimates are derived for the numerical solution. Numerical examples are provided in support of the proposed method.

A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then the domain of definition of the differential equation (a closed interval) is divided into three non-overlapping sub-intervals, which we call them inner regions (boundary layers) and outer region. Then the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newtons method of quasi-linearization is applied. The present method is demonstrated by providing examples. The method is easy to implement and suitable for parallel computing.

A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers

Singularly perturbed turning point problems (TPPs) for second order ordinary differential equations (DEs) exhibiting twin boundary layers are considered. In order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods. In this method, the given internal (domain of definition of differential equation) is divided into four subintervals and we solve the differential equation in each interval and combine the solution. The implementation of the method on parallel architectures is mentioned. Error estimates are derived and numerical examples are given.

An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations

In this paper a numerical method is suggested to solve a class of boundary value problems for a weakly coupled system of singularly perturbed second order ordinary differential equations of reaction-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of the well known perturbation method. Then initial value problems and terminal value problems (TVPs) are formulated such that their solutions are the terms of this asymptotic expansion. These problems are happened to be singularly perturbed problems and therefore exponentially fitted finite difference schemes are used to solve these problems. Since the boundary value problem is converted into a set of initial and TVPs and an asymptotic expansion approximation is used, the present method is termed as an asymptotic initial value method. Necessary error estimates are derived and examples provided to illustrate the method. The present method is easy to implement and well suited for parallel computing.

A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations

Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable initial and boundary conditions. Then the domain of definition of the differential equation (a closed interval) is divided into two sub-intervals, which we call the inner region (boundary layer) and the outer region. Then the DE is solved in these intervals separately. The solutions obtained in these intervals are combined to give the solution in the whole interval. To obtain boundary conditions at the transition points (boundary values inside this interval) we use mostly the zeroth-order asymptotic expansion of the solution of the BVP or a suitable asymptotic expansion solution. First, the linear equations are considered and then the semi-linear equations. To solve semi-linear equations Newtons method of quasi-linearisation is applied. Examples are provided to illustrate the method. The method is easy to implement and suitable for parallel computing.

A computational method for solving singular perturbation problems

Abstract A class of singularly perturbed two point boundary value problems for second order ordinary differential equations is considered. In order to solve them, a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods. The proposed method is distinguished by the following facts: first, we divide the given interval (the domain of definition of differential equation) into two subintervals called inner and outer regions. Then we solve the differential equation over both the regions as two point boundary value problems. The terminal boundary condition of the inner region is obtained using the zero-order asymptotic expansion of the solution. Some numerical examples are given to illustrate the method.

A numerical method for singular perturbation problems arising in chemical reactor theory

Abstract A class of singularly perturbed two point boundary value problems for second order ordinary differential equations with mixed boundary conditions, arising in chemical reactor theory is considered. In order to solve them, a numerical method is suggested, in which an exponentially fitted difference scheme is combined with classical numerical methods. The proposed method is distinguished by the following facts: first, we divide the given interval (the domain of definition of the differential equation) into two subintervals called outer and inner regions. Then, we solve the differential equation over both the regions as two point boundary value problems. The terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution. Some numerical examples are given to illustrate the method.

An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.

An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type☆

Abstract Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.

A “Booster method” for singular perturbation problems arising in chemical reactor theory

A numerical method for singularly perturbed two-point boundary-value problems (BVPs) for second-order ordinary differential equations (ODEs) arising in chemical reactor theory is proposed. In this method, an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution. Uniform error estimates are derived when implemented in known difference schemes. Numerical results are presented in support of this claim.